In Leonhard Euler's seminal work Introductio in Analysin Infinitorum (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the Introductio using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson's distinctive version of NSA. We will specifically examine Euler's proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic arguments. In some cases, the epsilontic arguments differ significantly from Euler's original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but certainly not all--of the inferential moves found in the Introductio.